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Problem 3

Let three circles Γ1,Γ2,Γ3\Gamma_1,\Gamma_2,\Gamma_3, which are non-overlapping and mutually external, be given in the plane. For each point PP in the plane, outside the three circles, construct six points A1,B1,A2,B2,A3,B3A_1,B_1,A_2,B_2,A_3,B_3 as follows: for each i=1,2,3i=1,2,3, Ai,BiA_i,B_i are distinct points on Γi\Gamma_i such that the lines PAiPA_i and PBiPB_i are both tangents to Γi\Gamma_i. Call PP exceptional if the three lines A1B1,A2B2,A3B3A_1B_1,A_2B_2,A_3B_3 are concurrent. Show that every exceptional point of the plane, if any exists, lies on the same circle.
Step 2 of 5: Place the contact chord on Ω
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Xi=POi∩AiBi,PXi⊥QXi⟹Xi∈ΩX_i=PO_i\cap A_iB_i,\quad PX_i\perp QX_i\Longrightarrow X_i\in\Omega
Detailed analysis

Let O_i and r_i be the center and radius of Γ_i, and let X_i=PO_i∩A_iB_i. The line A_iB_i is perpendicular to PO_i (the contact chord is the polar of P), so PX_i is perpendicular to QX_i. Hence X_i lies on the circle Ω with diameter PQ.